Two questions from Dugundji's book (not hw, just practice).
1) Let $Y_{1}, Y_{2}$ be subspaces of $X$ and $A \subset Y_{1} \cap Y_{2}$. Assume that $A$ is open in $Y_{1}$ and open in $Y_{2}$. Prove A is open in $Y_{1} \cup Y_{2}$.
Can you please give a hint for this one?
2) a. Let $D$ be dense in $X$. Give an example to show that $D \cap A$ need not to be dense in $A$.
Can't we take $X = \mathbb{R}$, $D=\mathbb{Q}$ then $D$ in dense in $X$. Now take $A =$ irrationals, since the empty set is closed then it cannot be dense in $A$.
b. If $A$ is dense in $B \subset X$ then $A$ is dense in $\overline{B}$.
Attempt:
Let $V \subset \overline{B}$ be an open set, then by definition of subspace topology we have $V = C \cap \overline{B}$ where $C$ is an open subset of $X$. Now consider $C \cap B$ ,this is an open set in $B$ so $A$ intersects this set and hence $A$ is dense in $\overline{B}$. What bothers me, is how do we know that $C \cap B$ is non-empty?
Thanks.
Best Answer
HINT For the first one Recall the definition of $A$ being open in the relative topology: if there exists some $A'$ open in $X$ such that $A'\cap Y_1 = A$.